1. Field of the Invention
The present invention relates to a water level regulating system including a plurality of water level regulators, each of which controls the water level of a head tank of individual hydroelectric plants installed in cascade along a water system.
2. Description of Related Art
A conventional water level regulator controls the water level of a head tank by the feedback control. The feedback control is carried out by a PID control using the opening of guide vanes of a hydraulic turbine as a manipulated variable, and the water level of the head tank as a controlled variable. Generally, the parameters of the PID control are fixed.
When a plurality of power plants are installed in cascade along a water system, each power plant is provided with a water level regulator that performs the PID control.
FIG. 1 shows a system which includes a plurality of tanks installed in cascade. In this figure, TK.sub.1 designates an upstream head tank, TK.sub.2 designates an upstream surge tank, TK.sub.3 designates a downstream head tank, TK.sub.4 designates a downstream surge tank, TK.sub.n designates an n-th tank, GV.sub.1 designates guide vanes of an upstream power plant, and GV.sub.2 designates guide vanes of a downstream power plant. In this system, a transfer characteristic of change in water level of each tank has nonlinear characteristics because the time constant of the transfer characteristic changes depending on the volume of water flowing through the water system.
This will be described in more detail. First, a basic characteristic equation indicating the relationship between the water level of a tank and the discharge (the rate of flow) is expressed as follows: ##EQU1## where Q.sub.i is the rate of inflow to a tank i, H.sub.i is the water level, and A.sub.i is the surface area of the tank i.
The rate of flow through a pipe is expressed by the following equation of motion. ##EQU2## where .gamma. is the density of fluid, S.sub.i is the cross-sectional area of a pipe i, L.sub.i is the length of the pipe i, V.sub.i is the velocity of flow in the pipe i, k.sub.i is the friction factor of the pipe i, and g is the acceleration of gravity.
Performing Laplace transform after substituting S.sub.i V.sub.i =Q.sub.i in equation (2), and expanding equation (2) into a Taylor series at a neighborhood of a reference value Q.sub.i0 of Q.sub.i, gives equation (3). ##EQU3##
The water level H.sub.1 (s) of the upstream head tank TK.sub.1 and the water level H.sub.3 (s) of the downstream head tank TK.sub.3 of FIG. 1 are expressed by equations (4) and (5) by taking the Laplace transform on equations (1) and (3), and then reducing them. ##EQU4##
Here, the values in equation (4) are expressed by equations (6)-(10), and the values in equation (5) are expressed by equations (11)-(15). ##EQU5##
In equations (6) and (7), Q.sub.20 is a reference flow rate of Q.sub.2, and L.sub.2 is a pipe length of a pipe associated with Q.sub.2, and in equations (11) and (12), Q.sub.40 is a reference flow rate of Q.sub.4, and L.sub.4 is a pipe length of a pipe associated with Q.sub.4.
From equations (4) , (6) and (7) , it is seen that the transfer characteristic between the water level H.sub.1 of the upstream head tank TK.sub.1 and the discharge Q.sub.3 through the guide vanes GV.sub.1 of the upstream power plant, or the transfer characteristic between the water level H.sub.1 and the discharge Q.sub.1 flowing into the upstream head tank TK.sub.1 changes with the discharge Q.sub.20 flowing from the upstream head tank TK.sub.1 to the surge tank TK.sub.2.
In addition, it is seen from equations (5), (11) and (12) that the transfer characteristic between the water level H.sub.3 of the downstream head tank TK.sub.3 and the discharge Q.sub.5 through the guide vanes GV.sub.2 of the downstream power plant, or the transfer characteristic between the water level H.sub.3 and the discharge Q.sub.3 through the guide vanes GV.sub.1 of the upstream power plant changes with the discharge Q.sub.40 flowing from the downstream head tank TK.sub.3 to the surge tank TK.sub.4.
This presents a problem in that it is difficult for a single stage conventional PID control to ensure a stable and well controlled result. In particular, it is difficult to achieve a high speed response to a change in the rate of inflow through an intake, or to changes in characteristics of the water system to be controlled, which are caused by changes in operation conditions of hydraulic turbines. Thus, stable control of the water levels of the head tanks was difficult.